Further details of Proposition REF  can be found in [1]}.
Note that the discretization (volume of \(q\)  and \(q^{\prime }\) ) plays a significant role in the transition probability bounds as the image of \(q\)  under the learned dynamics can be very large (especially when it is over-approximated).
In addition, the regression errors dictate both the \(P[\Vert \hat{f}^{(i)}- f^{(i)}\Vert ^{q,u}_\infty \le \epsilon _i]\)  terms along with affecting the magnitude of the expanded and reduced sets. Hence, the larger these errors, the looser the bounds \(\check{P}\)  and \(\hat{P}\) , leading to larger uncertainty in the abstraction.
